/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 When converted to an iterated in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 27

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. $$\iint_{R}(y+1) e^{x(y+1)} d A ; R=\\{(x, y): 0 \leq x \leq 1,-1 \leq y \leq 1\\}$$

Short Answer

Expert verified
Question: Determine the best order of integration for evaluating the double integral $$\int_{-1}^{1}\int_{0}^{1}(y+1)e^{x(y+1)}dxdy$$ and find its value. Answer: The best order of integration is to proceed with the given order. The value of the double integral is $$e^2 - 3$$.

Step by step solution

01

Integrand simplification

The integrand is \((y+1)e^{x(y+1)}\). There is no simplification possible before integrating.
02

Determine the best order of integration

The bounds are \(0 \leq x \leq 1\) and \(-1 \leq y \leq 1\). We have two possible orders of integration: $$\int_{-1}^{1}\int_{0}^{1}(y+1)e^{x(y+1)}dxdy$$ or $$\int_{0}^{1}\int_{-1}^{1}(y+1)e^{x(y+1)}dydx$$. Looking at the integrand, we don't have any clear advantage, so we will proceed with the given order.
03

Integrate with respect to x

$$\int_{0}^{1}(y+1)e^{x(y+1)}dx$$ To integrate with respect to x, we perform a substitution: let \(u = (y+1)x\), so \(du = (y+1)dx\). Now the integral becomes $$\int (y+1)e^{(y+1)x}d(u/(y+1))= \int e^u du$$. The bounds of \(x\) are now \(0\) and \((y+1)\): $$\int_{0}^{y+1} e^u du$$ Now integrate to get $$E(u) = e^u$$ Evaluate at the bounds: $$E(y+1) - E(0) = e^{y + 1} - e^0 = e^{y + 1} - 1$$
04

Integrate with respect to y

Now we integrate the result with respect to y: $$\int_{-1}^1 (e^{y + 1} - 1) dy$$ We split the integral into two parts: $$\int_{-1}^1 e^{y + 1} dy - \int_{-1}^1 dy$$ First integral: $$E'(y) = e^{y + 1}$$ Evaluate at the bounds: $$E'(1) - E'(-1) = e^{2} - e^0 = e^2 - 1$$ Second integral: $$Y(y) = y$$ Evaluate at the bounds: $$Y(1) - Y(-1) = 1 - (-1) = 2$$ Now we add the two results: $$\boxed{e^2 - 1 - 2 = e^2 - 3}$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use spherical coordinates to find the volume of the following solids. The solid cardioid of revolution \(D=\\{(\rho, \varphi, \theta): 0 \leq \rho \leq 1+\cos \varphi, 0 \leq \varphi \leq \pi, 0 \leq \theta \leq 2 \pi\\}\)

Let \(D\) be the solid bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a>0, b>0,\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=\)au, \(y=b v, z=c w\) Evaluate \(\iiint_{D}|x y z| d A\)

Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. A solid cone has a base with a radius of \(a\) and a height of \(h\). How far from the base is the center of mass?

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. The wedge cut from the cardioid cylinder \(r=1+\cos \theta\) by the planes \(z=2-x\) and \(z=x-2\)

Mass from density Find the mass of the following solids with the given density functions. Note that density is described by the function \(f\) to avoid confusion with the radial spherical coordinate \(\rho\). The ball of radius 8 centered at the origin with a density \(f(\rho, \varphi, \theta)=2 e^{-\rho^{3}}\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.